## Factor Label Review

For any equation, as long as you know the units, “factor labeling” can be used to solve the problem. Remember, this is basically like a puzzle. You are going from one unit to another until you get to the unit that is

How many meters are in 3.5 kilometers?

1 1 km

(Remember from math, if the same number or unit is over itself this equals 1; which is why the

Aluminum has a density of 2.70 g/cm³. If there is a block of aluminum that occupies 4.5cm3, how many grams is it?

1 1 cm³

*wanted*. Conversion factors and ratios are what allow you to go from one unit to another. Again, these are numeric relationships between different units.*Sample Problem 1*How many meters are in 3.5 kilometers?

- You have to go from kilometers to meters using a conversion factor. There is only one conversion factor: 1 km = 1000 m.
- Factor labeling (aka dimensional analysis) requires the canceling of units (i.e., cross canceling). This is done by orienting the conversion factor such that the
*given*unit is on the bottom …

__3.5 km__x__1000 m__= …and the unit*wanted*is on top.1 1 km

(Remember from math, if the same number or unit is over itself this equals 1; which is why the

**given**unit is cancelled)*Sample Problem 2*Aluminum has a density of 2.70 g/cm³. If there is a block of aluminum that occupies 4.5cm3, how many grams is it?

- On this problem, you have to go from cm³ (volume) to grams (mass) using a conversion factor. There is only one conversion factor that involves mass and volume: density.
- Just like before, cancel your units by orienting the conversion factor such that the
*given*unit is on the bottom …

__4.5____x____cm³____2.70 g__= …and the*wanted*unit is on the top.1 1 cm³

In some problems, more than one step may be required. Notice, though, that in each step, the unit

Sample Problem 3

33 mm = ___ hm

1 1 mm 100 m

**Notice here, that we are converting from one prefix to another prefix; so the middle step is converting to the base unit. The base unit is what both prefixes have in common.

Sample Problem 4

How many gallons are in 225 milliliters?

1 1 mL 3.785 L

One last point for converting squared or cubed units. The factor label steps do not change. You still place the unit

Sample Problem 5

3 cm³ = ___ in³

1 2.54 cm

**You should notice something: “cm” do not cancel “cm³ ”. So, you have to cube the unit to cancel it.

1 (2.54 cm)³

*given*is placed on the bottom and the unit*wanted*winds up on top.Sample Problem 3

33 mm = ___ hm

- You have to go from millimeters to meters, then from meters to hectometers (because each metric prefix has a numeric value; it means a number)
- Just as you have done, cross cancel the units by placing your
*wanted*unit on top and your*given*unit on the bottom.

__33 mm__x__0.001 m__x__1 hm__=1 1 mm 100 m

**Notice here, that we are converting from one prefix to another prefix; so the middle step is converting to the base unit. The base unit is what both prefixes have in common.

Sample Problem 4

How many gallons are in 225 milliliters?

- You have to identify the unit that gallons and milliliters have in common, i.e., liters. So our conversion factors are : 1mL = 0.0001 L ; 3.785 L = 1 gal
- Once again, our plan has not changed – the
*given*unit is placed on the bottom and the*wanted*unit is placed on top**in all steps**.

__225 mL__x__0.001 L__x__1 gal__=1 1 mL 3.785 L

One last point for converting squared or cubed units. The factor label steps do not change. You still place the unit

*given*on the bottom and the unit*wanted*on top. What is affected, however, are the numbers put in front of each unit.Sample Problem 5

3 cm³ = ___ in³

- You have to go from centimeters to inches
- Just as you have done, cross cancel the units by placing your
*wanted*unit on top and your*given*unit on the bottom.

__3 cm³__x__1 in__=1 2.54 cm

**You should notice something: “cm” do not cancel “cm³ ”. So, you have to cube the unit to cancel it.

**Y****ou must cube the number to cube the unit**.__3 cm³__x__(1 in)³__= This is always done, even if several steps are required.1 (2.54 cm)³